[seqfan] A242003
israel at math.ubc.ca
israel at math.ubc.ca
Thu Aug 14 00:54:26 CEST 2014
This is one of a number of similar recently-approved sequences from
Vaclav Kotesovec. It is titled "G.f. satisfies: A(x) = x + A(x*A(x))",
and with offset of 0 the data are given as 1, 1, 1, 3, .... But the g.f.
1 + x + x^2 + 3*x^3 + ... wouldn't satisfy the equation in the title.
It seems to satisfy 2*A(x) = 1 + x + A(x*A(x)) instead.
For a power series A(x) to satisfy A(x) = x + A(x*A(x)), the coefficient
a(0) can be anything except 1. The solution with a(0) = 0 would be
x+x^2+x^3+2*x^4+4*x^5+8*x^6+17*x^7+38*x^8+87*x^9+203*x^10+482*x^11+...
The sequence of coefficients here is A006196 (and the fact that the g.f.
of this sequence satisfies A(x) = x + A(x*A(x)) is noted there).
So, what to do with A242003?
I haven't checked the other sequences in this group. Someone should.
Cheers,
Robert
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